Question

Find the value of m for which the system of equations

x - 2y + z = 0,

-2x - y + 3z = 0

y + z = m 

has only trivial solution.

Answer 1

Since trivial solution is (x,y,z)=0, we have m=0.

The system is

$\left\{ \begin{array}{c} x-2y+z=0\\ -2x-y+3z=0\\ y+z=0\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} x-2y-y=0\\ -2x-y-3y=0\\ z=-y\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} x=3y\\ x=-2y\\ z=-y\\\end{array} \right. \Rightarrow x=y=z=0$

Thus m=0 is the answer.